M E C H A N I K A T E O R E T Y C Z N A I S T O S O W A N A
2/3, 21 (1983)
T H E ANALYSIS O F A VISCOELASTIC S H E L L O F R E V O L U T I O N PERIODICALLY L O A D E D P I O T R K O N D E R L A The shell of revolution periodically loaded by mechanical forces or by the periodically forced displacement is analysed in the paper. The material of a shell is assumed to be linear. The problem will be solved on the basis of the technical theory of thin shells [1]. Under assumption that the displacements and deformations are small the problem is linear. It is also assumed that the loadings of the shell (mechanical and nonmechanical) are described by the harmonic functions with the circular frequency с о and are applied in the duration of the sufficient length. It allows to assume that the motion of the construc tion is stationary, i.e. that the displacements and internal forces vary periodically with the frequency o>. It enables searching of the solution without explicit time function. The set of curvilinear coordinates л : = (f, rj) (Fig. 1) is chosen in the middle surface of the shell. F o r simplification it is assumed that all loads are in the same phase and that the origin of the time axis is chosen so, that the phase angle is equal to zero. It does not restrict the generality of considerations because the problem is linear and the principle of superposition holds. F o r to = 0 the statical problem has been received. Wroclaw 1. Introduction Fig. 1
170 P. K O N D K R L A The solution of the problem has been obtained numerically by Finite Element Method [2, 3]. The construction of the F E M algorithm for eight parameter conical elements are given in the paper. The solution formulated in this way has been applied to the analysis of the rubber construction of the seal of rotational shafts. • 2 . Basic definitions and linear viscoelasticity equations Let the system of cylindrical coordinates X = (r, z, rj) be introduced in the Euclidean space. The middle surface of the shell immersed in this space occupies the region & The curvilinear coordinates x = (f, tj), interrelated with (/, z, tj) by means of A' = [r(f), z(f), tj], parametrize the surface. The material of the shell is isotropic with mass density Q0 and thickness distribution Л (Ј). A l l functions which describe the motion, deformations and stresses in the shell will be described in terms of the physical coordinates of the two dimensional tensor fields on the middle surface of the shell. Keeping in mind the applied method of the solution we have assumed it in the form of vectors in the local set of coordinates (x, Ј). The motion of the middle surface is described by the displacement vector ( 1 ) f(x,t) = [u(x,t),v(x,t),w(x,t)]T . Moreover, the concept of the generalized vector of displacement is introduced (2) f(x, t) = [u(x, t), v{x, t), w(x, t), tptfx, t), (pn(x, f ) f , where cpc(x, t), q>n(x, t) are angles of rotations of the material fibre normal to the mid surface. The stresses and deformations in the shell are described by vectors ( 3 ) e(x, t) = [ee(x, t), en(x, t), y(x, t), xs(x, t), x„(x, t), tfx, t)f,
( 4 ) a(x, t) = [ns(x, t), nn(x, t), щ п(х , t), п ц {х , t), mr,(x, t), m^(x, /)].' Geometrical relations have the form, [2] (5) e(x,t)= PJ[f(x,t)], where is the linear differential operator of geometrical relations for thin shells o f revolution. Stressstrain relations will be written down in the form of the Voltera equation,
№
(6) a(x, t) = 3F*[e(x, t)]. For v = const E q . (6) can be rewritten in the form t (7) a{x,t) = D(i)\e(x,t) f l \ t т )е (х , т )с 1г \. оwhere Д г , т ) = JT(/— Т ) is the relaxation speed function, while 0 ( f ) is the elasticity matrix for the isotropic linear elastic body.
L E P K O S P R Ę Ż Y S TA P O W L O K Ą O B R O T O W A the energy. F o r isothermic problems we have (8) A ? + c ) = Z , where: К is the power of the kinetic energy (9) K& J ' [f(x,t)]T o(i)f(x,t)d&, t (10) p(l) = diag[o0, o0, go, ft 2 o0/12, й \ )0/ 1 2 ] , U is the power of the internal energy (11) U = J [i(x,t)]T o(x,t)dv, d and L is the power of the external forces (12) L= j [f(x,t)]T p(x,t)d&. 171 3. The algorithm of F E M Let the shell be divided into a number of the conical finite elements and the lines of nodes coincide with the chosen parallels [2]. The nodal parameters in the node a(a = 1 , 2 ) are represented by the vector (Fig. 2) (13) 'I Fig. 2 Taking into account the axial symmetry of the structure, we are to expand the nodal parameters into the Fourier series along the tj variable. If the motion is an effect of the periodical excitation with the frequency to, then (14) 1 = 0
172 P. KONDIRl Л where yl (rj) diag [c,. su Cj, ct] [cosh] Icosh] is °' \sinlrj ' 1 \cos//y for odd loads' <5i—real part and imaginary part (respectively) of the displacement vector o f the a node. Displacements o f eelement will be written down in the form (i5) fw (xlt \ 0 = Ke0e) )tie) (v 0 = Јм 0В А ф № Щ % = ^ 4 ^ № ( ^ i < ' V ' ' \ where: x( e ) = (Ј<«>,»j) z'(rj) = diag [ci, s,, с ,]. Table 1 sin/// for even loads 1f 0 0 0 0 0 0 0 l 3 Ј2 + 2Ј3 5 0
^ '
0 0 f 0 0 0 0 3f2 2f3 In E q . (15) typical base functions were used, namely linear for displacements ;/ and • v while cubic for the displacement w. The shape matrix N?ti is presented in Table I. Here and in what follows the summation convention holds. Similarly for the generalized vector of displacements (2) we have (16) Geometrical relations in accordance with (5) have the form (17) E« > ( X« \ о = / ) ] = a i ^ Ą ^ N ^ m d 1 * ^ ] = i where: г (rj) = diag [с ,, c,, s,, c,, c,,s{\,(2
+
М р + I • w" ++
+ I+
« л • I + I ++
I с м+
I >+
+ I+
| ч <N 1 *—\ 1 • MJI 1 о "§•1 »• 1 v | t i 1 o *N \» Г М • « N ^ \ JJ.fl |н 1i o o O
1 + Mr I + I
+
JJJ1 I ^ |N + m + I 4^* I4*
i • 4/. I4j>
+
w Mr - !«+
Э 1 <N I + JJJ. I I+
4* I [ П З )174 P. K O N U E R L A
B[c)(S<e)
) — geometrical matrix is presented in Table 2.
After the conversion of E q . (7) and substitution into E q . (17) we have
i (18) a<e >(x, t) = D ( P) ) [ f it e > ( x « ' \ t)f Д / т ) «( с) ( х ( с) , T)dr] = • • 0 *1 и м *ш • I = 0 ( ^ > )Ј< % х < е \ 0 ( Г с + /Г ,) = where . Z V = 1—/ F(z)cosojzdz, T\ = J A z ) s i n r o z d z . о о The shell as an assemblage of elements e = 1, 2, E occupies the region •& which is different from the region § . The total nodal parameters of the structure уj are related to the local parameters of the element by the transformation (19) where (20) (21) Л <е > for e + xl=j, 0 in the a l l remaining cases, coS(/)(L,) 0 sin(/>(e > 0" 0 1 0 0 sin</>(c) 0 c o s 0( e ) 0 0 0 0 1 Applying the transformation (19) to Eqs. (15), (16), (17) and (18) we arrive at the expressions describing the state of displacement, deformations and stresses at each point of the region v in the form E e=l I I (22) f(x,t) = z E A v v € Aw q e m = H'(V)^j ^)УWV ', e l I a(x,t) = (Г с + И \) V ^ ( f ' ^ ^ ^ W C ) ^ 0 ) ^ '0 ' ^ ' " ' m = ( г с+ и \ ) У D ( f)?(rj)&'(F)
aje*',
where x = ( f, rj) and | — is the coordinate which coincides with the generator of the element.Substituting R H S of Eqs. (22) into E q . (8) and the performing simple conversions, we obtain the equation of equilibrium of the structure in the form
L E P K O S P R Ę Ż Y S TA P O W Ł O K A O B R O T O W A 175 (23) (Tc+irs)K lin dl nco 2 Mlin ól n = F tJ , for / = 0 , 1 , 2 , j= 1 , 2 , . . . , Ј + 1 ; where (24) M y " = Jf [ z ^ ) V V W K Ј ) z 4 ^ m F) ^ , о Comparing real and imaginary parts o f E q . (23) we obtain the conjugatejsystem o f equations with real roots {rcK lin m2 Mlin )6l nrsK ,in 6l n = Щ (25) _ = _ rsK' jn d, n(rcK ,JN M2 MlJn )d, n = F lj , for / = 0, 1 , 2 , j = 1,2, . . . , E + 1 . Together with Eqs. (25) the suitable displacement boundary conditions have to be taken into account. 4. The numerical example
Forced vibrations of the shell o f revolution being the model o f the rubber construction of the seal of rotational shafts (Fig. 3) is analysed. The linear viscoelastic body, for which the velocity o f relaxation function takes the form (26) A / ) = C e ' V 1 , for t > 0. is assumed as the model o f the material. 0.65 cm I
176 P. K O N D I R L A The material constants were found on the basis of the laboratory tests: С = 0,236; fi = 0,01; x = 0,1; Youngs modulus of elasticity E = 9,4 M P a . The mass density of ma terial is equal to Q0 = 1250 k g / m 3 . The forced displacements of the structure are given by its loading as a result of the eccentric location of the axis of rotational shaft in the relation to the axis of rotation. The motion of the shaft can be decomposed into two simple harmonic motions i n two perpendicular directions, displaced in relation to each other by phase angle equal л /2. Dynamical thrusts of the seal lip on the shift are the most interesting values from the point of view of applications. O n F i g . 4 the plot of amplitudes of the unitary thrusts o f the seal lip as the function of angular velocity to is given. The motion of the structure in which different influences are taken into account has been analysed: a) the motion of the structure in which the mass of the spring and the friction are neglec ted, (line 1 on F i g . 4), b) the mo'tion of the structure with the influence of the mass of the spring and without the friction (line 2 on F i g . 4), c) the motion of the structure with the influence of the mass of the spring and the friction (line 3 on F i g . 4). pr[N/m] 1 i, /— r udird/s] 10 DO 2C Ю 0 V 3C Ю 0 3 X — > 2
\
XX
Fig. 4 In each case the forced displacements of the nodal line in the place of contact of the seal with the shaft are equal to Wj(r), t) = wjcos^e'0 " for wj = 0,01 cm, in the direction perpendicular to the axis of the shaft. It is assumed that the constraints in the place o f the contact are bilateral. The coefficient of friction ц was taken equal to 0,1. Line 4 on Fig. 4 shows unitary thrust as the result of the static axisymmetrical extension of the seal lip by value wf = 0,01 cm.LKPKOSPitĘ Ż YSTA POW LOKĄ OB R OTOW A 17" Literature
1 W . F L U G E , Powłoki, obliczenia statyczne. Arkady, Warszawa 1972.
2 О . C . Z I E N K I E W I C Z , Mitoda elementуw skoń czonych. Arkady, Warszawa, 1972.
3. J . T . O D E N , Finite elements of nonlinear continua, McGrawHill Book Company, New Y o r k . 1972. 4. J . N . R A B O T N O W , Eleimnty msłjedstwiennoj mechaniki twierdych tjel, Nauka, Moskwa, 1977. Р е з ю м е А Н А Л И З В Я З К О У П Р У Г О Й О Б О Л О Ч К И В Р А Щ Е Н И Я П Е Р И О Д И Ч Е С К И Н А Г Р У Ж Е Н Н О М В р а б о т е а н а л и з и р у ю т с я о б о л о ч к и в р а щ е н и я ц и к л и ч е с к и н а г р у ж е н н ы е м е х а н и ч е с к о й н а г р у з к о й и л и ц и к л и ч е с к и в ы н у ж д е н н ы м п е р е м е щ е н и е м . М о д е л ь м а т е р и а л а э т о т е л о л и н е й н о Е Я З 1« Т < у п р у г о е . П р о б л е м ы р е ш а ю т с я и с х о д я и з у р а в н е н и й т е х н и ч е с к о й т е о р и и т о н к и х о б о л о ч е к п о л ь з у я с ь м е т о д о м к о н е ч н ы х э л е м е н т о в . Д л я и л л ю с т р а ц и и а н а л и з и р у ю т с я в ы н у ж д е н н ы е к о л е б а н и я о б о л о ч к и в р а щ е н и я с о с т а в л я ю щ е й м о д е л ь к о н с т р у к ц и и р е з и н о в о г о у п л о т н е н и я в р а щ а н щ я х с я п а л о в . S t r e s z c z e n i e
A N A L I Z A L E P K O S P R Ę Ż Y S T EJ P O W Ł O K I O B R O T O W E J O B C I Ą Ż O N EJ P E R I O D Y C Z N I E W pracy analizowano p o w ł o k a o b r o t o w ą o b c i ą ż o ną cyklicznie o b c i ą ż e n i em mechanicznym lub Cyidicz nie wymuszonym przemieszczeniem. Modelem materiału jest c i a ł o liniowo l e p k o s p r ę ż y s t e. Zagadnienie r o z w i ą z a no w oparciu o zwią zki technicznej teorii p o w ł o k cienkich, p o s ł u g u j ą c się m e t o d ą e l e m s n t у w s k o ń c z o n y c h. Jako ilustrację analizowano drgania wymuszone p o w ł o k i obrotowej bę dą cej modelem konstrukcji gumowego uszczelnienia wirują cych w a ł у w .
